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arxiv: 2604.27670 · v1 · submitted 2026-04-30 · 🧮 math-ph · math.AP· math.DG· math.MP

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Hamilton--Jacobi theory for non-conservative field theories in the k-contact framework

Cristina Sard\'on, Javier de Lucas, Julia Lange, Xavier Rivas

Authors on Pith no claims yet

Pith reviewed 2026-05-07 05:19 UTC · model grok-4.3

classification 🧮 math-ph math.APmath.DGmath.MP
keywords Hamilton-Jacobi theoryk-contact geometrynon-conservative field theoriesdissipative systemsHamilton-De Donder-Weyl equationsevolution vector fieldsintegrable k-vector fields
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The pith

Non-conservative field theories admit two Hamilton-Jacobi formulations reconstructed from integrable k-vector fields in k-contact geometry.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper sets out to extend Hamilton-Jacobi theory to non-conservative classical field theories, with emphasis on dissipative systems, inside co-oriented k-contact geometry. It introduces evolution k-contact k-vector fields, writes the corresponding Hamilton-De Donder-Weyl equations, and constructs two families of Hamilton-Jacobi theories. One family reconstructs the dynamics from an integrable k-vector field on the base manifold of (⊕^k T^*Q)×R^k → Q; the other uses the base manifold of the same bundle over Q×R^k. Special attention is paid to Hamiltonians affine in the dissipative variables, quadratic dependence is shown to enlarge the range of models, and the ordinary contact theory is recovered for k=1 without some earlier technical restrictions. The resulting first-order PDEs are illustrated on the telegrapher equation, a dissipative Hunter-Saxton equation, a non-regular first-order model, and a relativistic thermodynamic system.

Core claim

Two distinct families of Hamilton-Jacobi theories are developed for non-conservative field theories in the k-contact setting. The z-independent approach reconstructs the dynamics from an integrable k-vector field defined on the base manifold of (⊕^k T^*Q)×R^k → Q. The z-dependent approach uses an integrable k-vector field on the base manifold of (⊕^k T^*Q)×R^k → Q×R^k. Explicit equations are obtained when the Hamiltonian depends affinely on the dissipative variables; quadratic dependence on those variables is used to enlarge the class of treatable systems. The construction recovers the standard contact Hamilton-Jacobi theory as the case k=1 while dropping previous technical assumptions.

What carries the argument

Evolution k-contact k-vector fields that satisfy the k-contact Hamilton-De Donder-Weyl equations and allow the dynamics to be reconstructed from integrable k-vector fields on the base manifolds of the bundle (⊕^k T^*Q)×R^k.

If this is right

  • The full second-order field equations of a dissipative system reduce to a first-order partial differential equation on the base manifold.
  • Affine dependence of the Hamiltonian on dissipative variables produces explicit, solvable Hamilton-Jacobi equations.
  • Quadratic dependence on dissipative variables systematically enlarges the set of models covered by the theory.
  • The k=1 limit recovers the ordinary contact Hamilton-Jacobi theory while removing prior technical restrictions.
  • Concrete examples such as the telegrapher equation and the dissipative Hunter-Saxton equation become solvable by the new first-order equations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same geometric reduction may supply variational principles or conserved quantities for other dissipative partial differential equations not treated in the examples.
  • Numerical schemes that integrate the reconstructed k-vector field could be tested on the affine and quadratic cases to compare accuracy with direct integration of the original field equations.
  • The framework suggests a systematic way to add controlled dissipation to multisymplectic field theories by lifting them into the k-contact bundle.

Load-bearing premise

The systems must admit a co-oriented k-contact structure whose dynamics can be reconstructed from integrable k-vector fields on the specified base manifolds.

What would settle it

A concrete dissipative field theory that possesses a co-oriented k-contact structure yet whose solutions cannot be recovered from any integrable k-vector field on either base manifold would falsify the two Hamilton-Jacobi constructions.

Figures

Figures reproduced from arXiv: 2604.27670 by Cristina Sard\'on, Javier de Lucas, Julia Lange, Xavier Rivas.

Figure 1
Figure 1. Figure 1: Infinitesimal element of a transmission line. view at source ↗
read the original abstract

This article develops a Hamilton--Jacobi theory for non-conservative classical field theories, with particular emphasis on dissipative systems, in the framework of co-oriented k-contact geometry. We introduce evolution k-contact k-vector fields, extending the contact evolution formalism to field theories, and analyse the corresponding Hamilton--De Donder--Weyl equations. Moreover, we develop two distinct families of Hamilton--Jacobi theories: a z-independent approach, based on the reconstruction of the dynamics from an integrable k-vector field defined on the base manifold of $(\bigoplus^kT^*Q)\times\mathbb{R}^k\to Q$, and a z-dependent approach, where the integrable k-vector field is defined on the base manifold of $(\bigoplus^kT^*Q)\times\mathbb{R}^k\to Q\times\mathbb{R}^k$. We develop in detail the important case of Hamiltonian functions with affine dependence on the dissipative variables, show how quadratic dependence on these variables can be used structurally to enlarge the range of applications, and recover the ordinary contact Hamilton--Jacobi theory as the particular case k=1, while removing some technical assumptions appearing in previous formulations. Our theory is illustrated through several representative examples, including the telegrapher/Klein--Gordon equation, a dissipative Hunter--Saxton equation, a simple dissipative non-regular first-order field model, and a relativistic thermodynamic model.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. This paper develops a Hamilton--Jacobi theory for non-conservative classical field theories in the co-oriented k-contact geometry framework. It introduces evolution k-contact k-vector fields, analyzes the associated Hamilton-De Donder-Weyl equations, and constructs two families of Hamilton--Jacobi theories: a z-independent approach based on reconstruction from an integrable k-vector field on the base of (⊕^k T^*Q)×R^k → Q, and a z-dependent approach using the base of (⊕^k T^*Q)×R^k → Q×R^k. Special emphasis is placed on the case of Hamiltonians with affine dependence on dissipative variables, with quadratic dependence discussed to enlarge the range of applications; the k=1 case recovers the ordinary contact Hamilton--Jacobi theory while removing some prior technical assumptions. The theory is illustrated with examples including the telegrapher/Klein--Gordon equation, dissipative Hunter--Saxton equation, a simple dissipative non-regular first-order field model, and a relativistic thermodynamic model.

Significance. If the central constructions and integrability conditions hold, this work is significant for extending geometric Hamilton--Jacobi methods to dissipative field theories, offering a structured approach to solving associated PDEs in non-conservative systems. The two distinct families provide flexibility in handling the dissipative variables, the explicit affine case treatment is useful for applications, and recovering the k=1 contact theory with relaxed assumptions strengthens the contribution. The examples from physics (wave equations with dissipation, thermodynamics) demonstrate relevance, and the reliance on standard co-oriented k-contact structures with case-by-case integrability checks aligns with established practices in geometric mechanics.

minor comments (3)
  1. In the section developing the Hamilton-De Donder-Weyl equations, the reconstruction of dynamics from the integrable evolution k-contact k-vector fields should include an explicit statement or lemma confirming that the integrability condition is preserved under the affine dependence on dissipative variables, as this is central to both HJ approaches.
  2. The abstract notes removal of technical assumptions from previous k=1 formulations; the introduction or the k=1 recovery subsection should explicitly list the removed assumptions (e.g., specific regularity or co-orientation conditions) and demonstrate how the new k-contact setup eliminates them.
  3. In the examples section, for each model (telegrapher/Klein--Gordon, dissipative Hunter--Saxton), add a brief verification that the chosen k-vector field satisfies the integrability condition on the specified base manifold; this would strengthen the claim that the framework applies without additional ad-hoc checks.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive and detailed summary of our manuscript, for recognizing its significance in extending Hamilton--Jacobi methods to dissipative field theories via the co-oriented k-contact framework, and for recommending minor revision. We are pleased that the referee highlights the value of the two families of theories (z-independent and z-dependent), the explicit treatment of affine and quadratic Hamiltonians, the recovery of the k=1 contact case with relaxed assumptions, and the physical examples. Since the report lists no specific major comments, we have no points requiring rebuttal or substantive revision at this stage. We will incorporate any minor editorial or presentational improvements in the revised version.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained in geometric definitions

full rationale

The paper constructs its Hamilton-Jacobi theory by first defining evolution k-contact k-vector fields as an extension of the standard co-oriented k-contact structure on the indicated bundles, then deriving the associated Hamilton-De Donder-Weyl equations directly from the contact form and the vector-field integrability condition. The z-independent and z-dependent HJ families are obtained by choosing integrable k-vector fields on the respective base manifolds (⊕^k T^*Q × R^k → Q and → Q × R^k), with the reconstruction of dynamics following from that integrability rather than being presupposed. The affine-dependence case is handled by explicit substitution into the same geometric equations, and the k=1 reduction is a direct specialization that removes prior technical assumptions without circular reference. Examples are presented only for verification, not for parameter fitting or self-validation. No load-bearing self-citations, self-definitional steps, or ansatzes imported via citation appear in the chain; all steps rest on independent differential-geometric constructions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claim rests on standard differential geometry axioms for contact structures and the domain assumption that integrable k-vector fields exist for dynamics reconstruction. The key invented entity is the evolution k-contact k-vector field introduced to extend prior formalisms. No free parameters are present as this is a pure theoretical development without data fitting.

axioms (2)
  • standard math Co-oriented k-contact manifolds admit the required contact forms and distributions for defining k-vector fields
    Invoked throughout the framework definition and when extending contact evolution to field theories.
  • domain assumption The dynamics can be reconstructed from an integrable k-vector field on the base manifold
    Central to both z-independent and z-dependent Hamilton-Jacobi approaches.
invented entities (1)
  • evolution k-contact k-vector fields no independent evidence
    purpose: Extend the contact evolution formalism to non-conservative field theories
    New object defined to analyze the corresponding Hamilton-De Donder-Weyl equations in the k-contact setting.

pith-pipeline@v0.9.0 · 5558 in / 1625 out tokens · 91629 ms · 2026-05-07T05:19:54.839937+00:00 · methodology

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